 
Summary: An Isoperimetric Inequality for the Heisenberg Groups
Daniel Allcock*
July 21, 1995; revised 5 February 1997
allcock@math.utah.edu
web page: http://www.math.utah.edu/¸allcock
Deptartment of Mathematics
University of Utah
Salt Lake City, UT 84112
1991 mathematics subject classification: 20F32 (53C20, 22E25)
Abstract.
We show that the Heisenberg groups H 2n+1 of dimension five and higher, considered as Rieman
nian manifolds, satisfy a quadratic isoperimetric inequality. (This means that each loop of length
L bounds a disk of area ¸ L 2 ). This implies several important results about isoperimetric inequal
ities for discrete groups that act either on H 2n+1 or on complex hyperbolic space, and provides
interesting examples in geometric group theory. The proof consists of explicit construction of a
disk spanning each loop in H 2n+1 .
1 Introduction
The Heisenberg groups H 3 ; H 5 ; H 7 ; : : : are a sequence of nilpotent Lie groups that arise in geom
etry in several ways. For example, H 3 is known to threedimensional geometers as Nilgeometry,
and arises in the study of Seifertfibered threemanifolds [14]. The H 2n+1 also appear in hyper
